IJMMS 2004:9, 469–478
PII. S0161171204203441
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR
INTEGRODIFFERENTIAL EQUATIONS
D. BAHUGUNA and REETA SHUKLA
Received 1 March 2001
We consider a class of quasilinear integrodifferential equations in a reflexive Banach space.
We apply the method of semidiscretization in time to establish the existence, uniqueness,
and continuous dependence on the initial data of strong solutions.
2000 Mathematics Subject Classification: 34G20, 47D06, 35L40, 35B30, 47H06.
1. Introduction. Let X and Y be two real reflexive Banach spaces such that the embedding Y X is dense and continuous. We consider the following quasilinear evolution equation:
u (t) + A u(t) u(t) = f t, u(t), G(u)(t) ,
0 < t ≤ T,
u(0) = u0 ,
(1.1)
where 0 < T < ∞, A(u) is a linear operator in X for each u in an open subset W of Y , G
is a nonlinear Volterra operator defined from C([0, T ]; X) into itself, and the nonlinear
map f is defined from [0, T ] × Y × Y into Y . By a strong solution to (1.1) on [0, T ],
0 < T ≤ T , we mean an absolutely continuous function u from [0, T ] into X such that
u(t) ∈ W for almost every t ∈ [0, T ] and satisfies (1.1) a.e. on [0, T ].
We use the ideas and techniques of Zeidler [10] and the method of semidiscretization
in time to establish existence, uniqueness, and continuous dependence on initial data
of strong solutions to (1.1) on [0, T ] for some 0 < T ≤ T . For the study of particular
cases of (1.1) in which f (t, u, v) ≡ 0 and f (t, u, v) ≡ f (t, u), we refer to Crandall and
Souganidis [2], Kato [6], and references cited therein. The crucial assumption in these
works is that there exists an open subset W of Y such that for each w ∈ W , A(w)
generates a C0 -semigroup in X, A(·) is locally Lipschitz continuous on W from X into
itself, f , defined from W into Y , is bounded and globally Lipschitz continuous from Y
into itself, and there exists an isometric isomorphism S : Y → X such that
SA(w)S −1 = A(w) + B(w),
(1.2)
where B(w) is in the space B(X) of all bounded linear operators from X into itself. A
generalization to the quasilinear evolution equation,
u (t) + A t, u(t) u(t) = f t, u(t) ,
u(0) = u0 ,
0 < t ≤ T,
(1.3)
470
D. BAHUGUNA AND R. SHUKLA
has been considered by Katō [4] under similar conditions on A(t, w) and f (t, w) for
(t, w) ∈ [0, T ] × W .
The method of semidiscretization in time is developed and applied to linear as well
as nonlinear evolution equations by Rektorys [9], Kartsatos and Zigler [3], Nečas [7],
Bahuguna and Raghavendra [1], and others. This method consists in replacing the time
derivatives in an evolution equation by the corresponding difference quotients giving
rise to a system of time-independent operator equations. With the help of the theory
of semigroups and the theory of monotone operators, these systems are guaranteed to
have unique solutions. An approximate solution to the evolution equation is defined
in terms of the solutions of these time-independent systems. After proving a priori
estimates for the approximate solution, the convergence of the approximate solution
to the unique solution of the evolution equation is established. In these works, either
global Lipschitz conditions or local Lipschitz conditions with some growth conditions
on nonlinear forcing terms have been assumed.
In this paper, we assume only local Lipschitz conditions on the nonlinear maps f and
G. We first prove that the discrete points lie in a ball in X of fixed radius R, where R is
independent of the discretization parameters. Then using the local Lipschitz continuity,
we establish a priori estimates on the difference quotients. With the help of these a
priori estimates, we prove the convergence of a sequence of approximate solutions
defined in terms of the discrete points to a unique solution of the problem.
2. Preliminaries. Let X and Y be as in Section 1. Let xZ denote the norm of an
element x belonging to a Banach space Z. For r > 0, let BZ (x, r ) denote the open ball
{z ∈ Z : z − xZ < r } of radius r and let B̄Z (x, r ) be its closure. For an interval J
of real numbers, we denote by C(J; Z), WC(J; Z), Lip(J; Z), and ABS(J; Z) the spaces
of all continuous, weakly continuous, Lipschitz continuous, and absolutely continuous
functions from J into Z, respectively.
For a real β, N(Z, β) represents the set of all densely defined linear operators L in
Z such that if λ > 0 and λβ < 1, then (I + λL) is one-to-one with a bounded inverse
defined everywhere on Z and
(I + λL)−1 B(Z)
≤ (1 + λβ)−1 ,
(2.1)
where I is the identity operator on Z. The Hille-Yosida (cf. Pazy [8]) theorem states that
L ∈ N(Z, β) if and only if −L is the infinitesimal generator of a strongly continuous
semigroup e−tL , t ≥ 0, on Z satisfying e−tL B(Z) ≤ eβt , t ≥ 0. A linear operator L on
D(L) ⊆ Z into Z is said to be accretive in Z if for every u ∈ D(L),
Lu, u∗ ≥ 0
for some u∗ ∈ F (u),
(2.2)
∗
where z, z∗ is the value of z∗ ∈ Z ∗ at z ∈ Z and F : Z → 2Z is the duality map given
by
2 F (z) = z∗ ∈ Z ∗ : z, z∗ = z2Z = z∗ Z ∗ .
(2.3)
METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR . . .
471
∗
Here, 2Z denotes the power set of Z ∗ . If L ∈ N(Z, β), then (L+βI) is m-accretive in Z,
that is, (L+βI) is accretive and the range R(L+λI) = Z for some λ > β. If Z ∗ is uniformly
convex, then F is single-valued and uniformly continuous on bounded subsets of Z.
(H) We assume in addition that the embedding Y X is compact and the dual X ∗ is
uniformly convex. Furthermore, we state the following hypotheses:
(H1) there exist an open subset W of Y and β ≥ 0 such that u0 ∈ W and
A : W → N(X, β);
(2.4)
(H2) there exist positive constants µA and γA such that for all w, w1 , w2 ∈ W and
v ∈ Y,
Y ⊆ D A(w) ,
A w1 − A w2 v ≤ µA w1 − w2 vY ,
X
X
A(w)v ≤ γA vY ;
X
(2.5)
(H3) there exist a linear isometric isomorphism S : Y → X, a map P : W → B(X), and
positive constants µP and γP such that for all w, w1 , w2 ∈ W ,
P (w) ≤ γP ,
SA(w) = A(w)S + P (w)S,
X
P w1 − P w2 ≤ µP w1 − w2 ;
X
Y
(2.6)
(H4) the nonlinear map G : C([0, T ]; X) → C(0, T ; X) satisfies
(a) for all u, v ∈ B̄C([0,T ];X) (ũ0 , r ),
G(u) − G(v)
C([0,T ];X)
≤ µG (r )u − vC([0,T ];X) ,
(2.7)
where µG (r ) is a nonnegative nondecreasing function and ũ0 ∈ C([0, T ]; X)
is defined by ũ0 (t) = u0 for all t ∈ [0, T ],
(b) for all t, s ∈ [0, T ] and u ∈ Lip([0, T ]; X) ∩ B̄C([0,T ];X) (ũ0 , r ),
G(u)(t) − G(u)(s) ≤ γG (r )|t − s| 1 + du/dtL∞ ([0,T ];X) ,
X
(2.8)
where γG (r ) is a nonnegative nondecreasing function,
(c) furthermore that the subspace C([0, T ]; Y ) of space (C[0, T ]; X) is an invariant subspace of the map G, that is, G : C([0, T ]; Y ) → C([0, T ]; Y ) which satisfies
G(u)(t) ≤ λG (r )
Y
for u ∈ BY u0 , r ,
(2.9)
where λG (r ) is a nonnegative nondecreasing function.
In particular, we may take operator G as a Volterra operator defined by
G(u)(t) =
t
0
a(t − s)k s, u(s) ds
(2.10)
472
D. BAHUGUNA AND R. SHUKLA
in which a is a real-valued continuous function defined on [0, T ] and k is defined on
[0, T ] × Y into Y and k(t, w)Y ≤ Ck for every (t, w) ∈ [0, T ] × Y , then map G satisfies
hypothesis (c);
(H5) the nonlinear map f : [0, T ] × Y × Y → Y is a bounded function
f (t, u, v) ≤ λf (r )
Y
(2.11)
for all (t, u, v) ∈ [0, T ]×Y ×Y with uY +vY ≤ r where λf (r ) is a nonnegative
nondecreasing function. Also, this map satisfies the Lipschitz condition
f t1 , u1 , v1 − f t2 , u2 , v2 X
≤ µf (r ) φ t1 − φ t2 + u1 − u2 X + v1 − v2 X ,
(2.12)
for all t1 , t2 ∈ [0, T ] and all ui , vi ∈ B̄X (u0 , r ), i = 1, 2, where φ is a real-valued
continuous function of bounded variation on [0, T ] and µf (r ) is a nonnegative
nondecreasing function.
Let R > 0 be such that WR = B̄Y (u0 , R) ⊆ W . We set
−1
R
1 + e2θT
,
3
R1 = max R, λG (3R) + u0 Y ,
M = λf R + u0 + λG (3R) ,
R0 =
(2.13)
where θ = β + P B(X) and V (φ) is the total variation of φ on [0, T ].
Let z0 ∈ Y and let T0 , 0 < T0 ≤ T , be such that
Su0 − z0 ≤ R0 ,
X
T 0 γ A z 0 Y + γ P z 0 X + M ≤ R 0 .
(2.14)
We note that (2.14) implies that
1 + e2θT
Su0 − z0 + T0 γA z0 + γP z0 + M ≤ 2R .
Y
X
3
(2.15)
We have the following main result for the existence, uniqueness, and continuous
dependence on the initial data of the strong solutions to (1.1).
Theorem 2.1. Let (H), (H1), (H2), (H3), (H4), and (H5) hold. Then there exists a unique
strong solution u to (1.1) on [0, T0 ] such that u ∈ Lip([0, T0 ]; X). Furthermore, if v0 ∈
B̄Y (u0 , R0 ), then there exists a strong solution v to (1.1) on [0, T0 ] with the initial point
u0 replaced by v0 and
u(t) − v(t) ≤ C u0 − v0 ,
X
X
where C is a positive constant depending only on T0 .
t ∈ 0, T0 ,
(2.16)
473
METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR . . .
3. Basic lemmas. We will state and prove several lemmas required to prove Theorem
2.1. A proof of Theorem 2.1 will be given in the next section.
Let w0 = Su0 . Let h = T0 /n for all positive integers n ≥ N where N is a positive
n
n
integer such that θ(T0 /N) < 1/2. For n ≥ N, we set un
0 = u0 , ũ0 = ũ0 , and tj = jh for
j = 1, 2, . . . , n. We consider the scheme
n n
n n
n n δun
,
j + A uj−1 uj = f tj , uj−1 , G ũj−1 tj
j = 1, 2, . . . , n,
(3.1)
where, for j = 1, 2, . . . , n, n ≥ N,
δun
j =
n
un
j − uj−1
h


u0 ,





n
1 n
n
ũn
t − ti−1
ui − un
j (t) = ui−1 +
i−1 ,

h



 un ,
j
,
t = 0,
n
, tin , i = 1, 2, . . . , j,
t ∈ ti−1
t ∈ tjn , T0 .
(3.2)
The following result establishes the fact that un
j ∈ WR , j = 1, 2, . . . , n, n ≥ N.
Lemma 3.1. For each n ≥ N, there exist unique un
j ∈ WR , j = 1, 2, . . . , n, satisfying
(3.1).
Proof. It follows (cf. [2, Lemma 2]) that there exists a unique un
1 ∈ Y such that
n
n
n un
.
1 + hA u0 u1 = u0 + hf t1 , u0 , G ũ0 t1
(3.3)
Applying S on both sides in (3.3) using (H3) and putting w1n = Sun
1 , we have
n
w1 − z0 + hA u0 w1n − z0 + hP u0 w1n − z0
= w0 − z0 − hA u0 z0 + hP u0 z0 + hSf t1n , u0 , G ũ0 t1n .
(3.4)
The estimates in [2, Lemma 2] imply that
n
w − z0 ≤ (1 − hθ)−1 w0 − z0 + h γA z0 + γP z0 + M .
1
X
Y
X
(3.5)
Since hθ < 1/2, we have
n
w − z0 ≤ e2θh w0 − z0 + h γA z0 + γP z0 + M .
1
X
Y
X
(3.6)
Therefore,
n
w − w0 ≤ 1 + he2θh w0 − z0 + h γA z0 + γP z0 + M ≤ R
1
X
Y
X
(3.7)
474
D. BAHUGUNA AND R. SHUKLA
n
in view of the estimate (2.15). Hence, un
1 ∈ WR . Now suppose that ui ∈ WR for i =
1, 2, . . . , j − 1. Again, [2, Lemma 2] implies that for 2 ≤ j ≤ n there exist unique un
j ∈Y
such that
n n
n n
n n n
.
un
j + hA uj−1 uj = uj−1 + hf tj , uj−1 , G ũj−1 tj
(3.8)
Proceeding as before and putting wjn = Sun
j , we get the estimate
n
w − z0 ≤ e2θh wj−1 − z0 + h γA z0 + γP z0 + M .
X
X
Y
X
j
(3.9)
Reiterating the above inequality, we get
n
w − z0 ≤ e2θjh w0 − z0 + jh γA z0 + γP z0 + M .
X
X
Y
X
j
(3.10)
Using the fact that jh ≤ T0 , we arrive at
n
w − w0 ≤ 1 + e2θT w0 − z0 + T0 γA z0 + γP z0 + M ≤ R.
X
X
Y
X
j
(3.11)
The above estimate and (3.3) and (3.8) give the required result. This completes the
proof.
Lemma 3.2. There exists a positive constant C independent of j, h, and n such that
δuj ≤ C,
X
j = 1, 2, . . . , n; n ≥ N.
(3.12)
Proof. Putting j = 1 in (3.1), we get
n n
n .
δun
1 + hA u0 δu1 = −A u0 u0 − f t1 , u0 , G ũ0 t1
(3.13)
Using [2, Lemma 2], we have
n
δu ≤ e2θT γA u0 + M := C0 .
1 X
Y
(3.14)
From (3.1), for 2 ≤ j ≤ n, we have
n n δun
j + hA uj−1 δuj
n n n
= δun
j−1 − A uj−1 − A uj−2 uj−1
n
n n n n − f tj−1
+ f tjn , un
, un
j−1 , G ũj−1 tj
j−2 , G ũj−2 tj−1 .
(3.15)
Using (H2) and [2, Lemma 2], for 2 ≤ j ≤ n, we get
n
δu ≤ e2θh 1 + C1 h δun j X
j−1 X
n n n n (3.16)
n
+ f tjn , un
, un
− f tj−1
X
j−1 , G ũj−1 tj
j−2 , G ũj−2 tj−1
METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR . . .
475
for some positive constant C1 independent of j, h, and n. We note that
n n n n G ũ
X
j−1 tj − G ũj−2 tj−1
n
δu ,
≤ µG (3R)hδun
j−1 X + γG (3R)h 1 + max
i X
1≤i≤j−1
n n
n n n n
f t , u , G ũn
, un
− f tj−1
X
j
j−1
j−1 tj
j−2 , G ũj−2 tj−1
n n
n ≤ µf R1 φ tj − φ tj−1 + hδuj−1 X + µG (3R)hδun
j−1 X
.
+ γG (3R)h 1 + max δun
i X
(3.17)
1≤i≤j−1
Using (3.17) in (3.16), we obtain
2θh
1 + C2 h
max δun
i X ≤e
1≤i≤j
n
n + C2 h ,
max δun
i X + C2 φ tj − φ tj−1
1≤i≤j−1
(3.18)
where C2 is another positive constant independent of j, h, and n. Reiterating inequality
(3.18) and using (3.14), we get
j 2θjh
max δun
1 + C1 h δun
1 X + C2 V (φ) + C2 T0 ,
i X ≤e
1≤i≤j
(3.19)
where V (φ) is the total variation of φ. Hence,
n
δu ≤ e2(θ+c2 )T c0 + c2 V (φ) + C2 T0 := C.
i X
(3.20)
This completes the proof.
Now we define a sequence of functions {U n } from J0 into Y by
U n (t) = un
j−1 +
1 n
t − tj−1 un
j − uj−1 ,
h
n
, tjn , j = 1, 2, . . . , n.
t ∈ tj−1
(3.21)
Furthermore, we define a sequence of step functions {X n } from (−h, T0 ] into Y given
by
n
X (t) =

 u0 ,
u ,
j
t ∈ (−h, 0],
n
t ∈ tj−1
, tjn , j = 1, 2, . . . , n.
(3.22)
Remark 3.3. We observe that X n (t) ∈ WR for all t ∈ (−h, T0 ] and n ≥ N. Also,
X n (t) − U n (t) → 0 in X uniformly on J0 as n → ∞ and {U n } are in Lip(J0 , X) with
uniform Lipschitz constant C.
For notational convenience, we will denote
n n f n (t) = f tjn , un
,
j−1 , G ũj−1 tj
n
, tjn , j = 1, 2, . . . , n.
t ∈ tj−1
(3.23)
476
D. BAHUGUNA AND R. SHUKLA
We note that
t
0
A X n (s − h) X n (s)ds = u0 − U n (t) +
t
o
f n (s)ds,
d
U n (t) + A X n (t − h) X n (t) = f n (t).
dt
−
(3.24)
(3.25)
Lemma 3.4. There exist a subsequence {U m } of {U n } and a function u in Lip(J0 , X)
such that U m → u in C(J0 , X) (with the supremum norm) as m → ∞.
Proof. Since {X n } is uniformly bounded in Y , the compact embedding of Y implies
that there exist a subsequence {X m } of {X n } and a function u : J0 → X such that
X m (t) → u(t) in X as m → ∞. The reflexivity of Y implies that u(t) is the weak limit of
X m in Y , hence u(t) lies in Y (in fact in WR since X m (t) is in WR ). Now, X m (t)−U m (t) →
0 in X so U m (t) → u(t) as m → ∞. The uniform continuity of {U n } on J0 implies that
{X m } is an equicontinuous family in C(J0 , X) and the strong convergence of U m (t) to
u(t) in X implies that {U m } is relatively compact in X. We use the Ascoli-Arzelá theorem
to conclude that U m → u in C(J0 , X) as m → ∞. Since U m are in Lip(J0 , X) with uniform
Lipschitz constant, u ∈ Lip(J0 , X). This completes the proof of the lemma.
4. Proof of Theorem 2.1. First we show that A(X m (t − h))X m (t) A(u(t))u(t) in
X as m → ∞, where “” denotes the weak convergence in X,
A X m (t − h) X m (t) − A u(t) u(t)
= A X m (t − h) − A u(t) X m (t) + A u(t) X m (t) − u(t) .
(4.1)
Now,
m
A X (t − h) − A u(t) X m X
m
≤ µA R + u0 Y X (t − h) − u(t)X → 0
(4.2)
as m → ∞, since X m (t) → u(t) in X uniformly on J0 . Since A(u(t)) ∈ N(X, β), βI +A(u)
is m-accretive in X. We use [5, Lemma 2.5] and the fact that
A u(t) X m (t) − u(t) ≤ 2γA R
X
(4.3)
to assert that A(u(t))X m (t) A(u(t))u(t) in X, and hence A(X m (t − h))X m (t) A(u(t))u(t) in X as m → ∞. To show that A(u(t))u(t) is weakly continuous on J0 ,
let {tk } ⊂ J0 be a sequence such that tk → t as k → ∞. Then u(tk ) → u(t) in X as
k → ∞ and we may follow the same arguments as above to prove that A(u(tk ))u(tk ) A(u(t))u(t) in X as k → ∞. The Bochner integrability of A(u(t))u(t) can be established
in the similar way as in Kato [5, Lemma 4.6]. Now from (3.24), for each x ∗ ∈ X ∗ , we
have
U m (t), x ∗ = u0 , x ∗ +
t
0
− A X m (s − h) X m (s) + f m (s), x ∗ ds.
(4.4)
METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR . . .
477
Letting m → ∞ using bounded convergence theorem and Lemma 3.4, we get
u(t), x
∗
= u0 , x ∗ +
t
0
− A u(s) u(s) + f s, u(s), G(u)(s) , x ∗ ds.
(4.5)
The continuity of the integrand implies that u(t), x ∗ is continuously differentiable
on J0 . The Bochner integrability of A(u(t))u(t) implies that the strong derivative of
u(t) exists a.e. on J0 and
u (t) + A u(t) u(t) = f t, u(t), G(u)(t) ,
a.e. on J0 .
(4.6)
Since u(0) = u0 , u is a strong solution to (1.1).
Now, we establish the uniqueness and the continuous dependence on the initial data
of a strong solution to (1.1).
Uniqueness. Let v be another strong solution to (1.1) on J0 . Let U = u − v. Then
for a.e. t ∈ J0 ,
dU
(t), F U (t) + βI + A u(t) U(t), F U(t)
dt
2 = βU (t)X + A u(t) − A v(t) v(t), F U(t)
+ f t, u(t), G(u)(t) − f t, v(t), G(v)(t) , F U(t) .
(4.7)
Using m-accretivity of βI +A(w) and the assumptions on A(w) for w ∈ W and f (t, u, v),
we obtain
1 d
U (t)2 ≤ CR U2
C(Jt ,X) ,
2 dt
(4.8)
where Jt = [0, t] and CT = β + µA R + µf (R)[1 + µG (R)].
Integrating the above inequality on (0, t) and taking the supremum, we get
1
U 2C(Jt ,X) ≤ CR
2
t
0
U2C(Js ,X) ds.
(4.9)
Applying Gronwall’s inequality, we get U ≡ 0 on J0 .
Continuous dependence. Let v0 ∈ BY (u0 , R0 ). Then
Sv0 − z0 ≤ Sv0 − Su0 + Su0 − z0 ≤ 2R0 .
X
X
X
(4.10)
Hence,
1 + e2θT Sv0 − z0 + T0 γA z0 Y + γP z0 X + M
≤ 3 1 + e2θT R0 = R.
(4.11)
We may proceed as before to prove the existence of vjn ∈ WR satisfying scheme (1.1)
n
n
with un
j and u0 replaced by vj and v0 , respectively. Convergence of vj to v(t) can be
proved in a similar manner. Let U = u − v. Then following the steps used to prove the
478
D. BAHUGUNA AND R. SHUKLA
uniqueness, we have for a.e. t ∈ J0 ,
1 d
U (t)2 ≤ CR U2
X
C(Jt ,X) .
2 dt
(4.12)
Integrating the above inequality on (0, t) and taking the supremum, we get
2
1
1
U 2C(Jt ,X) ≤ U (0)X + CR
2
2
t
0
U 2C(Js ,X) ds.
(4.13)
Applying Gronwall’s inequality, we get
U C(Jt ,X) ≤ C U(0)X ,
(4.14)
where C is a positive constant. This proves the required result. This completes the proof
of Theorem 2.1.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
D. Bahuguna and V. Raghavendra, Rothe’s method to parabolic integro-differential equations via abstract integro-differential equations, Appl. Anal. 33 (1989), no. 3-4, 153–
167.
M. G. Crandall and P. E. Souganidis, Convergence of difference approximations of quasilinear evolution equations, Nonlinear Anal. 10 (1986), no. 5, 425–445.
A. G. Kartsatos and W. R. Zigler, Rothe’s method and weak solutions of perturbed evolution
equations in reflexive Banach spaces, Math. Ann. 219 (1976), no. 2, 159–166.
S. Katō, Nonhomogeneous quasilinear evolution equations in Banach spaces, Nonlinear
Anal. 9 (1985), no. 10, 1061–1071.
T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–
520.
, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974), Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin, 1975, pp. 25–70.
J. Nečas, Application of Rothe’s method to abstract parabolic equations, Czechoslovak Math.
J. 24(99) (1974), 496–500.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,
Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and Its Applications (East European Series), vol. 4, D. Reidel Publishing,
Dordrecht, 1982.
E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B, Springer-Verlag, New
York, 1990.
D. Bahuguna: Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur 208
016, India
E-mail address: dhiren@iitk.ac.in
Reeta Shukla: Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur 208
016, India
E-mail address: reetas@lycos.com